On the soft limit of tree-level string amplitudes

0.12 MB·

by  Massimo Bianchi

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On the soft limit of tree-level
string amplitudes
MassimoBianchiandAndreaL.Guerrieri∗
Dipartimento di Fisica and INFN, Universit di Roma “TorVergata”,
Via della Ricerca Scientifica, 00133 Roma, Italy
∗E-mail: [email protected]
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1 We study the softbehavior of string scattering amplitudes at three level withmassless
0 and massive external insertions, relying on different techniques to compute 4-points
2 amplitudesrespectivelywithopenorclosedstrings.

n
Keywords:soft-theorems;higher-spins;stringperturbationtheory.
a
J

4 1. General introduction
1
The study of the soft limit of scattering amplitudes is a well established topic in

] QFT since the pioneering works of Low1 in quantum electrodynamics, and Wein-
h
t berg2, Gross and Jackiw3 in Einstein gravity. Recently, this subject has raised
-
p renewed attention due to the discovery4 of its connection with the B2MS large
e diffeomorphisms invariance of general relativity for asymptotically flat spaces5–12.
h
Soft theorems have been derived for a wide class of QFTs and strings from the
[
examination of scattering amplitudes at tree and loop level13–22. Concretely, if we
1
consider color-ordered gluon amplitudes, the soft theorem relates the n+1 ampli-
v
7 tude n+1(k1,...,kn;kn+1) to the n-points one n(k1,...,kn) through a differential
A A
5 operatordependingonthemomentumandthepolarizationofthe(n+1)-thexternal
4 leg, when k =δkˆ and δ 0; in formulae
3 n+1 n+1 →
0 a k a k f :J f :J
n+1 1 n+1 n n+1 1 n+1 n
1. An+1(k1,...,kn;kn+1)=(cid:18)k k −k k +k k −k k (cid:19)An(k1,...,kn),
n+1 1 n+1 n n+1 1 n+1 n
0 (1)
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up to terms of order (δ0). In Eq. (1) we introduced the field-strength f =
1 O µν
: aµkν aνkµ, and the angular-momentumoperatorJµν. A strongerstatement holds
v −
for gravity amplitudes: the soft behavior is “universal” up to terms of order (δ),
i O
X instead of order (δ0) as for the Yang-Mills case
O
r
a (k ,...,k ;k )=
n+1 1 n n+1
M
n
k h k k h J k k J h J k
i n+1 i i n+1 i n+1 n+1 i n+1 i n+1
+ + (k ,...,k ). (2)
n 1 n
(cid:18) k k k k 2k k (cid:19)M
Xi=1 i n+1 i n+1 i n+1
Foropensuperstringandbosonicstringtheories,Eq.(1)describesthesoftbehavior
of massless string amplitudes, notwithstanding the presence of an interaction term
α′F3 inthebosoniccase,thatinprinciplecouldspoilthevalidityofEq.(1). Closed
superstring amplitudes satisfy Eq. (2), while closed bosonic strings don’t because
at order (δ) there is a mixing between the graviton and the dilation, due to the
O
interaction term α′ϕR2.
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2. Soft theorems in QFT and strings
In this section we sketchin some detail the proof of Eq. (1) for color-orderedgluon
amplitudes in QFT – see Ref.13 for a more complete discussion, and compare this
approach to the one relying on the OPE expansion presented in Ref.14.
2.1. Gluon amplitudes in QFT
The singular soft behavior of the tree level color ordered amplitude
(k ,...,k ;k ),whenk becomessoft,arisesfromthe propagatorsinthechan-
n+1 1 n s s
A
nels k +k and k +k , in formulae
1 s n s
(k ,...,k ;k )=
n+1 1 n s
A
V (k ,k ) V (k ,k )
µ s 1 µ(k +k ,...,k )+ µ n s µ(k ,...,k +k )+R (k ,...,k ;k ).
(k +k )2 An 1 s n (k +k )2 An 1 n s n+1 1 n s
s 1 s n
(3)
The Yang-Mills vertex for incoming momenta k ,k ,k , with k = k k , can be
s i I I i s
− −
written as
V (k ,k )=V(0)+V(1) =2aµa k +2kµa a 2aµa k , (4)
µ s i µ,i µ,i i s i s s i− s i s
∂
V(0)=V (0,k )=2aµa k and V(1)=k V (0,k )=2kµa a 2aµa k .
µ,i µ i i s i µ,i s∂k µ i s s i− s i s
s
(5)
The expansion of the n-point amplitude around k yields
n i
A
∂
µ(...,k +k ,...)= µ(0)+ µ(1)+...= µ(...,k ,...)+k µ(...,kˆ,...) +...,
An s i An,i An,i An i s∂kˆ An i |kˆi=ki
i
(6)
Combining it with Eq. (4) produces the full soft expansion of Eq. (3). Imposing
gauge invariance order by order in the soft parameter δ, we can determine the
leading term of the unknown contribution R (k ,...,k ;k ). At leading order
n+1 1 n s
Vµ(,01) µ(0) Vµ(,0n) µ(0) = ask1 askn (k ,...,k )as→ks 0. (7)
2k k An,1 −2k k An,n (cid:18)k k − k k (cid:19)An 1 n −→
s 1 s n s 1 s n
At sub-leading order, it is possible to recognize a manifest gauge invariant contri-
bution
Vµ(,11) µ(0) Vµ(,1n) µ(0)=fsµνa1[ν∂/∂aµ1] (0) fsµνan[ν∂/∂aµn] (0) as→ks 0, (8)
2k k An,1 − 2k k An,n 2k k An − 2k k An −→
s 1 s n s 1 s n
whichreconstructstheactionofthespinpartoftheangularmomentum,andanon
gauge invariant part
V(0) V(0)
µ,1 µ(1) µ,n µ(1)+a R (k ,...,k ;k =0)
2k k An,1 −2k k An,n s n+1 1 n s
s 1 s n
as→ks k ∂ k ∂ +e k R (k =0)=0. (9)
s n s n s n+1 s
−→ ∂k A − ∂k A
1 n
e
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Solving Eq. (9) for R (k =0) allows us to recover the action of the orbital part
n+1 s
of the angular-momentum.
2.2. Gluon amplitudes in string theory
Theaboveanalysis,performedforaQFT,canbereproducedaswellbytheOPEof
vertexoperatorsinstringtheory. Tocomputeandn+1superstringgluonamplitude,
weneedtwooperatorsintheq= 1super-ghostpictureandn 2verticeswithq=0.
− −
Letussupposethesoftvertexoperatortobeintegratedandintheq=0super-ghost
picture, and the adjacent operators to be in the q= 1 picture. The leading order
−
in k , when z approaches z , is given by
s s s+1
zs+11
dz a i∂XeiksX(z )a ψe−ϕeiks+1X(z )...
s s s s+1 s+1
Z ∼
zs+11 a k
dz a k (z z )ksks+1... s s+1... (10)
s s s+1 s s+1
Z − ≈ ksks+1
Taking the sub-leading order in the soft expansion of Eq. (10), it happens that
BRST invarianceis responsiblefor reproducingthe actionofthe orbitalpartofthe
angularmomentum, as well as gaugeinvariancedoes in Eq.(9)14. The OPEof the
manifestly BRST invariant sub-leading term 1ψµf ψνeikX reproduces the action
2 µν
of the spin partof the soft operator. The above analysis is remarkable because can
be easily generalized when the OPE is performed between the soft operator and a
massive higher-spin string belonging to the leading Regge trajectory.
2.3. Soft behavior of disk scattering amplitudes with massive states
In the open bosonic string, at the first massive level, we have only one BRST-
invariant state given by the vertex operator
V =H i∂Xµi∂XνeipX (11)
H µν
with pµH =0, symmetric and traceless. Conversely, in the open superstring we
µν
have two different states interpolating in the q= 1 picture by
−
V =H i∂Xµψνe−ϕeipX and V =C ψµψνψρe−ϕeipX, (12)
H µν C µνρ
withpµC =0andcompletelyantisymmetric. InRef.24 wehaveshownthatstring
µνρ
disk-amplitudes with the insertions of the aforementioned vertex operators satisfy
Eq. (1). Following the steps outlined in Sec. 2.2, it is straightforwardto show that
also amplitudes with the insertion of massive higher-spin strings like
VHs =Hµ1...µsi∂Xµ1...i∂Xµs−1ψµse−ϕeipX (13)
satisfy the soft theorem.
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3. Massive string amplitudes from Yang-Mills
In Ref.23 it is shown that open string disk-amplitudes can be expressed in terms
ofYang-Millstree-amplitudes coveredby suitablegeneralizedhypergeometricfunc-
tions
ST(1,...,n)= F(1,[2 ,...,(n 2) ],n 1,n) YM(1,[2 ,...,(n 2) ],n 1,n).
An σ − σ − An σ − σ −
σ∈XSn−3
(14)
For n=5 Eq. (14) yields
ST(12345)=F(1[23]45) YM(1[23]45)+F(1[32]45) YM(1[32]45). (15)
A5 A5 A5
AswesuggestedinRef24,itispossibletoget4-pointamplitudeswithmassivestates
taking the residue at the massive pole in some 2-particle channels of the 5-points
amplitude with massless strings
lim (s +1) ST(1,2,...,n)= ST(1,2,S′) (S′,3,...,n). (16)
s12→−1 12 An XS′ A3 ⊗An−1
3.1. Dimensional reduction at D = 4
Before showing an explicit example, let us briefly review the content of the first
massive level of the superstring theory in the bosonic sector. The first massive
super-multiplet in D =10 yields a long multiplet of =4 in D =4
N
H ,C H ,8ψ ,27Z ,48χ,42ϕ . (17)
MN MNP µν µ µ
{ } → { }
The bosonic degrees of freedom are related to the 10-dimensional vertex operators
as follows24
H H 27Z 6H ,15C ,6C 42ϕ 21H ,20C ,C .
µν µν µ µ,i µ,ij µν,i ij ijk µνρ
← ← ←
(18)
Among these states, only three of them are R-symmetry singlets in D = 4: the
spin 2H ,thepseudo-scalarC =ε pσCµνρ/M andthescalarH comingfrom
µν 0 µνρσ 0
−
the fact that the BRST-invariant string state H is traceless only in D=10,
MN
thereforeafterdimensionalreductionweareleftwiththepartialtraceH =H δ /2.
ij 0 ij
3.2. Massive string amplitudes in D = 4
InD =4thereareonlyMHVoranti-MHV5-pointsgluonamplitudes. Forexample,
fixing the helicity of the gluons, Eq. (15) yields
12 3 12 4
ST(1−2−3+4+5+)= h i F(1[23]45)+ h i F(1[32]45).
A5 23 34 45 51 13 32 24 45 51
h ih ih ih i h ih ih ih ih i
(19)
Taking the residue at the pole s +1, we get
12
lim (s +1) ST(1−2−3+4+5+)= (1 α′s,1 α′t)
s12→−1 12 A5 B − − ×
12 3 12 4
h i [s s +(s +s )s ] h i s s . (20)
23 35 34 35 24 13 24
23 34 45 51 − 13 32 24 45 51
h ih ih ih i h ih ih ih ih i
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Itisremarkabletonoticethatatotallysymmetricandtracelessstate,asH ,orany
µν
otherspin sstateS couplesonlytovectorbosonswithoppositehelicity,let’s
− µ1...µs
say f± and f∓. Conversely, a scalar state couples only to vector bosons with the
1 2
same helicity f± and f±. This can be easily understood looking at the properties
1 2
of the ’t Hooft symbols which form a basis for the selfdual or anti-selfdual 4 4
×
matrices
(f+) (f−)νρ ηa η¯bνρ =Sabρ (21)
1 µν 2 ∝ µν µ
(f+) (f+)νρ ηa ηbνρ =δabδρ +εabcηc ρ. (22)
1 µν 2 ∝ µν µ µ
This observation allows us to disentangle the contribution due to the polarization
H++ or H−− from those due to the propagationof H or C 24
0 0
ST(1−2−3+4+5+)s12→−1 h12i2 (1 α′s,1 α′t) M[35] C /H (23)
A5 −→ M ×B − − 34 45 0 0
h ih i
ST(1+2+3−4−5+)s12→−1 [12]2 (1 α′s,1 α′t)h34i3[35] C /H (24)
A5 −→ M ×B − − M3 45 0 0
h i
13 4[35]
ST(1+2−3−4+5+)s12→−1M (1 α′s,1 α′t) h i H++/H−−.
A5 −→ ×B − − M 12 2 34 45
h i h ih i
(25)
Using SO(3) little group transformations that leave unchanged the momentum of
the massive particle p = uu¯+vv¯, it is straightforward to get the expression of the
amplitude for all the other polarizations of the tensor H. Defining
1 1
L : u′ = (u+v) v′ = ( u+v) (26)
x
√2 √2 −
1 1
L : u′ = (u+iv) v′ = (iu+v), (27)
y
√2 √2
and noticing that H+++H−− Lx−Ly H0, H++ H−− Lx (H− H+)/2, and
−→ − −→ −
H++ H−− Ly i(H−+H+)/2, we get the full amplitude rotating the polariza-
− −→
tions using linear combinations of the operators in Eq. (27)
c (1−2+3+Hh)= (1 α′s,1 α′t)
h
A B − −
Xh
[13] 14 2 15 2 14 2 14 2 14 2 14 2 14 2
h i h i c h i 4c h i +6c h i 4c h i +c h i .
× M 12 23 45 2 (cid:18) ++ 15 2− +0 15 2 00 15 2− 0− 15 2 −− 15 2(cid:19)
h ih ih i h i h i h i h i h i
(28)
4. Closed superstring amplitudes
We conclude with some few comments about the soft behavior of closed string
amplitudes with gravitonsandmassivestates studiedin detailin Ref.25. Using the
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amplitudes computed in Ref.24, and the KLT formula26
α′t
( , , , + + )=sin π (1,2,3,H +C ) (1,3,2,H +C ).
4 1 2 3 4 4 4 4L 4 4 4R 4 4
M E E E K L U (cid:18) 4 (cid:19)A ⊗A
(29)
Wecheckedthecorrectsoftbehavioruptothesub-sub-leadingorderinD =4using
the spinor-helicity formalism for amplitudes involving gravitons, dilatons, and the
massive states and
K H
(1−2,2+2,3+2, +4) (10,20,3+2, +4) (10,2+2,3+2, +2), (30)
M4 K4 M4 K4 M4 H4
when the graviton with momentum k becomes soft. While for bosonic closed
3
string amplitudes we have found a discrepancy at sub-sub-leading order due to α′
dependent terms in agreement with Ref.27.
Acknowledgments
We wish to thank A. Addazi, D. Consoli, P. Di Vecchia, Y. Huang, R. Marotta, F.
Morales,L. Pieri,O.Schlotterer,S.Stieberger,G.Veneziano andC.Wenfor many
precious discussions and comments.
References
1. F. E. Low, Phys. Rev. 96, 1428 (1954).
2. S. Weinberg, Phys. Rev. 140, B516 (1965).
3. D. J. Gross and R. Jackiw, Phys. Rev. 166, 1287 (1968).
4. T. He, V. Lysov, P. Mitra and A. Strominger, JHEP 1505, 151 (2015)
[arXiv:1401.7026[hep-th]].
5. H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, Proc. Roy. Soc. Lond.
A 269, 21 (1962).
6. G. Barnich and C. Troessaert, Phys. Rev. Lett. 105, 111103 (2010)
[arXiv:0909.2617[gr-qc]];PoS,010(2010)[Ann.U.CraiovaPhys.21,S11(2011)]
[arXiv:1102.4632[gr-qc]]; JHEP 1112, 105 (2011) [arXiv:1106.0213[hep-th]].
7. A. Strominger, JHEP 1407, 152 (2014) [arXiv:1312.2229[hep-th]].
8. F. Cachazo and A. Strominger, [arXiv:1404.4091[hep-th]].
9. S. W. Hawking, M. J. Perry and A. Strominger, [arXiv:1601.00921[hep-th]].
10. D.Kapec,V.Lysov,S.PasterskiandA.Strominger,[arXiv:1502.07644[gr-qc]].
11. S. Pasterski, A. Strominger and A. Zhiboedov, [arXiv:1502.06120[hep-th]].
12. A. Strominger and A. Zhiboedov, [arXiv:1411.5745[hep-th]].
13. Z.Bern,S.Davies,P.Di VecchiaandJ.Nohle,Phys.Rev.D90,no.8,084035
(2014) [arXiv:1406.6987[hep-th]].
14. M. Bianchi, S. He, Y. t. Huang and C. Wen, Phys. Rev. D 92, no. 6, 065022
(2015) [arXiv:1406.5155[hep-th]].
15. A. Brandhuber, E. Hughes, B. Spence and G. Travaglini, [arXiv:1511.06716
[hep-th]].
January15,2016 1:18 WSPCProceedings-9.75inx6.5in main page7
7
16. H. Luo and C. Wen, [arXiv:1512.06801[hep-th]].
17. Y. t. Huang and C. Wen, JHEP 1512, 143 (2015) [arXiv:1509.07840[hep-th]].
18. A. Volovich, C. Wen and M. Zlotnikov, JHEP 1507, 095 (2015)
[arXiv:1504.05559[hep-th]].
19. W.M.Chen,Y.t.HuangandC.Wen,JHEP1503,150(2015)[arXiv:1412.1811
[hep-th]]; Phys.Rev. Lett. 115, no. 2, 021603(2015)[arXiv:1412.1809[hep-th]].
20. S. G. Avery and B. U. W. Schwab, [arXiv:1512.02657 [hep-th]];
[arXiv:1510.07038 [hep-th]]; Phys. Rev. D 93, no. 2, 026003 (2016)
[arXiv:1506.05789[hep-th]].
21. B. U. W. Schwab, JHEP 1503, 140 (2015) [arXiv:1411.6661 [hep-th]]; JHEP
1408, 062 (2014) [arXiv:1406.4172[hep-th]].
22. I. Low, [arXiv:1512.01232[hep-th]].
23. C. R. Mafra, O. Schlotterer and S. Stieberger, Nucl. Phys. B 873, 419 (2013)
[arXiv:1106.2645[hep-th]];Nucl.Phys.B873,461(2013)[arXiv:1106.2646[hep-
-th]].
24. M. Bianchi and A. L. Guerrieri, JHEP 1509, 164 (2015) [arXiv:1505.05854
[hep-th]]; A. L. Guerrieri, [arXiv:1507.08829[hep-th]].
25. M. Bianchi and A. L. Guerrieri, [arXiv:1512.00803[hep-th]].
26. H. Kawai, D. C. Lewellen and S. H. H. Tye, Nucl. Phys. B 269, 1 (1986).
27. P. Di Vecchia, R. Marotta and M. Mojaza, JHEP 1505, 137 (2015)
[arXiv:1502.05258 [hep-th]]; P. Di Vecchia, R. Marotta and M. Mojaza,
[arXiv:1511.04921[hep-th]];P.DiVecchia,R.Marotta,M.MojazaandJ.Nohle;
[arXiv:1512.03316[hep-th]].